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Logic

Ex1

 

 

 

 

 

What reasonable conclusion can you make?

A : All Lower 6 students take Liberal Studies.
B : You are wrong.

All Lower 6 students do not take Liberal Studies.
Some Lower 6 students take Liberal Studies.
Some Lower 6 students do not take Liberal Studies.

Ex2

 

 

 

 

 

 

 

 

 

 

 

Conditionals "if p, then q"

Determine the truth value (true/false) for each of the following statements.

         
Truth value
  If p, then q.  
p
q
If p, then q
(i) If 2 is even, then 20 is odd.  
T
F
?
(ii) If 1 = 2, then 3 = 3.*  
F
T
?
(iii) If 1 = 2, then 11 = 22.**  
F
F
?
           
  Possible explanations for        
*(ii) If 1 = 2, then 2 = 1. Adding the two, 1+2=2+1, 3=3!      
**(iii) If 1 = 2, then 10=20. Adding the two, 11=22!      
         
  An example for (iii) in daily life: If a question in the examination is wrong, even if the candidate makes a logically sound argument, the candidate probably will reach a wrong conclusion.    

 

 

Ex3

(a)

 

 

 

The box is cubic in shape and it is red in colour. Negation of this statement is
The box is not cubic in shape and it is not red in colour.
The box is not cubic in shape or it is not red in colour.
The box is not cubic in shape or it is red in colour.

Guess a formula for the negation of "p and q".

(b)

 

 

 

 

The printer is not working. The connecting cable to the computer is loose or the printer is out of order. Negation of this statement is
The connecting cable to the computer is not loose or the printer is not out of order.
The connecting cable to the computer is not loose and the printer is out of order.
The connecting cable to the computer is not loose and the printer is not out of order.

Guess a formula for the negation of "p or q".

(c)(i)

 

 

 

 

 

 

If it is a Tuesday, then John needs to go to school. Negation of this statement is

(Hint: Note that "it is a Tuesday" is given. Only the conclusion "John needs to go to school" should be discussed. )

If it is a Tuesday, then John does not need to go to school.
If today is not Tuesday, then John does not need to go to school.
It is a Tuesday and John does not need to go to school.

(c)(ii)

 

 

 

 

Amy reached the final round of the TV game show. Before her were 3 keys from which she would choose one to open the door that leaded to the grand prize. Negation of the statement "If Amy chooses key #2, then she will win" is (Again note that "Amy chooses key #2" is a given fact, discuss only the conclusion.)
If Amy chooses key #2, then she will not win.
Amy chooses key#2 and she does not win.
Amy does not choose key#2 and she will win.

 
Can you guess a formula for the negation of "If p, then q"?
Go back to the notes and look up the correct formulae for the
negation of (a) p and q (b) p or q (c) If p, then q.

Do they agree with your intution?Now prove these formulae mathematically by completing the truth tables of (a) and (b).

You will find these formulae very useful in logical deduction, in mathematics as well as in daily life experience. Try Ex 4 using these formulae.

Ex5 There are 3 balls in a box.

(a)

 

 

 

All the balls are blue in colour. Negation of the statement is
All the balls are not blue in colour.
Some balls are blue in colour.
Some balls are not blue in colour.

(b)

 

 

 

Some balls are red in colour. Negation of the statement is
All the balls are not red.
All the balls are red.
Some balls are not red.
  Go back to the notes and try Ex 5 and Ex6.
Ex6 Hints for Ex6(a) and (*) are given below.

(a)

 

 

Statement: All mathematics problems are easy.
Negation of the statement :
in symbol: equivalent to
in words : Some mathematics problems are not easy.
Also,
   
Ex7(a) Consider the obvious fact "If ABCD is a square, then ABCD is a parallelogram".
  What words are missing below? Check your answer by clicking on the buttons.
 
(a) "ABCD is a square" is a ________condition for "ABCD is a parallelogram"
(b)"ABCD is a parallelogram" is a ________condition for "ABCD is a square"
(c)ABCD is a square _____ ABCD is a parallelogram.
  Go back to the notes and try Ex7(b).
   
Ex8

Prove that "If x2 is not an even integer, then x is not an even integer."

  Given that x2 is not an even integer,  
  To show that x is not an even integer.  
Pf Assume that x is an even integer.  
  It follows that x2 is an even integer.  
  But it is given that x2 is not an even integer!!! Contradiction!
  The assumption must be incorrect. Hence x is not an even integer.
   
Ex9 Prove that is irrational.
Pf Assume thatis rational.  
  Let =p/q where p and q are integers with no common factor other than 1 and q is not equal to 0.  
  By squaring and rearranging terms, we have p2 = 2 q2 .  
  Since p2 is even, it follows that p is even.  
  Let p = 2m where m is an integer. Subsituting back into the equation, we have (2m)2 = 2 q2 . On further simplification, 2 m2 = q2 .  
  Since q2 is even, it follows that q is even.  
  As p and q are both even, then 2 is a common factor of p and q!!! Contradiction!  
  The assumption must be incorrect. Hence is irrational.  

Problem solving

  A power point file on problem solving strategies and presentation techniques
 

 

Calculus

 

Functions with very special properties

(1) A plane filling curve - e.g. Peano curve

(2) A function which is not continuous at any point - Dirichlet function

(2) A continuous but nowhere differentiable function - Every differentiable function is continuous on IR. The converse is not true. There exist continuous functions on IR which is not necessarily differentable on IR. Actually, there exists a continuous function which is nowhere differentiable(e.g. Weiserstrass function).

Websites  
Calculus Toolkit MathServe Project by Vanderbilt University, USA offers useful calculations
  Limits;Differentiation;Integration;Sum of a series;Partial Fractions;Finding roots...
Curve Sketching

by Vanderbilt University, USA

  sketches curves y= f(x), Parametric Functions,Implicit Functions
  eg y = sin(3*x)+(cos(2*x))/6, y = 4*x^2 - 3*x + 6
Function Plotter by Maths Online, University of Vienna
Famous Curves by University of St. Andrews, Scotland
Thinkquest library extensive material on differentiation and integration
Java Applets excellent applets byInernational Education Software
Calculus recommended sites by Math Forum

 

e and the natural logarithm
  Application

(1)

 

 

It is known that for the same principal deposited in a bank, the more frequent the compound interest is credited the more one gets in the end. In other words, daily compounding yields more interest than monthly compounding which in turn yields more than yearly compunding. If a bank offers continuous compounding, is it as good as it sounds? Does it mean that the total amount will increase without bound?

(a)

 

 

 

Compare the amount in a bank account at the end of 10 years for a deposit of $P at an interest rate of 6 % compounded (a) yearly (b) half-yearly (c) monthly (d) daily (e) continuously. Answer

In general, find the total amount in the bank account for a deposit of $P at an interest rate r% compounded continuously over n years.

(Hint: assume the interest is compounded n times a year and then take n to infinity. Note that no matter how frequent interest is compounded, the total amount is bounded by the limit value Per%t)

(b)

 

 

 

 

 

The rule of 70:

Suppose it takes n years for an amount of money to double when invested at the rate of r% compounded continuously. The rule of thumb states that nr is approximately 70.

The proof is quite short. Try it.

In fact, even if the amount is compounded annually, the rule of thumb still works provided that r% is small compared to 1. Try proving it! (You may use the approximation that ln (1+x) is close to x when x is small.)

 

(2) Franklin Benjamin's will
  Franklin Benjamin (1706-1790) American printer, author, diplomat, philosopher, and scientist, inventor of the lighting rod and bifocal glasses
  Summary
 

 

 

 

 

Franklin Benjamin would give 1000 pounds to Boston (and another 1000 to Philadelphia). The plan was to lend money to young apprentices in these cities at 5% interest with the provision that each borrower should return the interest and part of the principal each year. Franklin predicted that if the plan was exceuted without interruption, the sum would reach 131,000 pounds at the end of 100 years, of which 100,000 pounds were to be allocated to public works in Boston and the remaining 31,000 pounds would continue to be lent to young people in the same manner for another 100 years. He predicted that if there was no unfortunate accident to prevent the operation, the sum would be 4,610,000 pounds.
  Fact
 

 

Though it was not always possible to find as many borrowers in Boston as Franklin had planned, the managers of the trust did the best they could. At the end of 100 years from the reception of the Franklin gift, in January 1894, the fund had grown from 1000 pounds to 90,000 pounds.
  Question

 

 

In the first 100 years since the will, the original capital had mutiplied about 90 times instead of 131 times Franklin had imagined. What rate of interest, compounded continuously, would have multipled the capital by 90?
    (Answer: 4.5%)
 

 

System of Linear Equations

  Amy, Ben and Calvin play a game as follows. The player who loses each round must give each of the other players as much money as the player has at that time. In round 1, Amy loses and gives Ben and Calvin as much money as they each have. In round 2, Ben loses, and gives Amy and Calvin as much money as they each then have. Calvin loses in round 3 and gives Amy and Ben as much money as they each have. They decide to quit at this point and discover that they each have $24. How much money did they each start with?
Sol This problem can be solved using the (1) top down strategy or (2) bottom up strategy
  Method 1
 
Amy
Ben
Calvin
At start
$x
$y
$z
after round 1
after round 2      
after round 3      

By completing the above table, you should obtain the following equations

x - y - z = 6, 3y - x - z = 12 and 7z - x - y = 24; from which you can solve for x, y and z. Answer

Linear Equation Solver

 

Method 2

Instead of considering how much money Amy, Ben and Calvin had originally, work backwards from the moment when each of them has $24 each. Complete the following table and see how easily you can reach exactly the same conclusion as in method 1.

 
Amy
Ben
Calvin
At the end
$24
$24
$24
before round 3
before round 2      
before round 1      

Matrices

 

Powerpoint files:

Matrix multiplication, cofactor and inverse

Examples on linear transformations

Partial Fractions

 

Thinkquest library - excellent website created by pre-collegiate students

 

Complex numbers

 

Harry found in his attic the old treasure map his great grandma inherited a long long time ago. His great grandma cherished the map over all these years and kept it a secret until she thought Harry was wise enough to recover the treasure and let the whole family enjoy the great wealth.

Sail to 114.5o North, 23.1oEast, there lies a small island. At the south of the island is a pasture where an oak tree and a pine tree and a gallows stand. Start walking from the gallows to the oak tree, counting carefully the number of steps taken. At the oak tree, turn right through 90 degrees and walk exactly the same number of steps. Make a mark there (P). Go back to th gallows and walk towards the pine tree, again counting the number of steps needed. From the pine tree, turn left through 90 degrees and again walk exactly the same number of steps. Make a mark there (Q). Start digging midway between P and Q. There lies the treasure.

Unfortunately, the wood gallows had vanished completely after all these years. On the other hand, the oak and the pine tree have survived numerous storms and become landmarks of the island. "Why didn't great grandma tell me earlier about the map? I could have easily found the treasure back then!" With great disappointment, Harry threw the map into the fire.

  This is a sad story. It is made even sadder by the fact that has Harry known more about mathematics (notably about complex numbers), he would have easily calculated the exact location of the treasure. If only he has paid more attention in class!
Sol

Imagine that the island lies on the Argand plane. The line joining the oak and the pine is the real axis. Without loss of generality, let O be the midpoint of the two trees and the trees' location are represented by the numbers 1 and -1 respectively. The gallows, P and Q are represented by z, p and q in the Argand plane.

Can you use the geometry of complex numbers to find p, q and hence z? Answer

 

 

 

Ans

(a)P(1+0.06)^10 (b)P(1+0.03)^20 (c)P(1+0.005)^120 (d)P(1+0.06/365)^3650

(e)limit value of P(1+0.06/n)^10n as n tends to infinity, on simplifying, this is Pe^0.6. Compare the results of (a) and (e). BACK

Ans

Solving the equations, Amy has $39, Ben $21, Calvin $12 before the game starts. BACK

Ans

p = i(z+1) + 1, q = i(1-z)-1, z = i !

In this case, Harry could have measured the distance between the oak and the pine trees (say x m) and then from midway of the two trees, he should walk in the direction making exactly a right angle with the line joining the two trees for x/2 m. There lies the treasure. BACK